A certain spring found not to obey Hooke's law exerts a...

A certain spring found not to obey Hooke's law exerts a restoring force $F_{x}(x)=-\alpha x-\beta x^{2}$ if it is stretched or compressed, where $\alpha=60.0 \mathrm{~N} / \mathrm{m}$ and $\beta=18.0 \mathrm{~N} / \mathrm{m}^{2}$. The mass of the spring is negligible. (a) Calculate the potential-energy function $U(x)$ for this spring. Let $U=0$ when $x=0 .$ (b) An object with mass $0.900 \mathrm{~kg}$ on a frictionless, horizontal surface is attached to this spring, pulled a distance $1.00 \mathrm{~m}$ to the right (the $+x$ -direction) to stretch the spring, and released. What is the speed of the object when it is $0.50 \mathrm{~m}$ to the right of the $x=0$ equilibrium position?

Key Concepts

Nonlinear Spring Force

This concept refers to a spring force that does not follow Hooke's law, meaning the restoring force is not strictly proportional to the displacement. Instead, additional terms, such as quadratic ones, are included to account for nonlinearity. Understanding nonlinear forces is important in advanced mechanics as many real-world systems exhibit deviations from simple linear behavior.

Potential Energy Function

For conservative forces, the potential energy function represents the stored energy as a function of position and is obtained by integrating the force. Specifically, since force is the negative gradient of the potential energy, U(x) can be determined by integrating the negative of the force with respect to displacement, which encapsulates the energy characteristics of the system.

Integration of Force

This key concept involves computing the potential energy by integrating the force function with respect to displacement. In conservative systems, the work done by a force is stored as potential energy, and this process links the differential force law to an energy formulation that is crucial for predicting system behavior.

Energy Conservation

The principle of conservation of energy states that in a closed system, the total energy remains constant if only conservative forces are at play. When a system involves potential energy from elastic forces, any change in potential energy is mirrored by a change in kinetic energy. This principle is fundamental in solving problems where dynamics are determined by the conversion between kinetic and potential energy.

Transcript

00:02 Hi, in the given problem, restoring force developed in the spring is variable, and that is given as a function of position, means displacement, means compression of the spring, fx, is equal to minus alpha x minus beta x square.

00:29 Here the values of alpha and beta are for alpha this is 60 newton per meter.

00:39 So this is minus 60 x and for beta this is 18 newton per meter squared.

00:47 So here it becomes minus 18 x square.

00:51 So first of all in the first part of the problem we have to obtain potential energy function using.

01:00 That function of the force restoring force.

01:04 So to find it we use a relation between the functions of potential energy and force which is given as f is equal to minus d u by d x.

01:20 So using that potential energy function will be given by the integration of the function of this force suppose from the limits of the distance from mean position when the object was at its mean position to any instantaneous position x minus hence it will come out to be negative of minus 60 x minus 18 x squared d x having the limits of x from 0 to x so taking this minus as a common out, it becomes integration of 60x dx from 0 to x and plus 18 x squared dx again from 0 to x.

02:19 Using the formula for integration, finally this potential energy function will come out to be 60 integration of x with respect to dx will be will become x square by 2 limits from 0 to x and plus 18 by 18 into x q by 3 having the same limits from 0 to x so this is given as 60 by 2 x x cube minus 0.

03:04 So finally this is 30 x squared plus 6 x cube.

03:12 And here it becomes the answer for the first part of the problem.

03:21 Now in the second part of the problem, mass of the object which is oscillating with the help of this spring is given as m is equal to 0 .9000 kilogram.

03:37 It's extreme.

03:39 Position when the speed becomes to 0 is given as 1 .00 meter...

Please give Ace some feedback